This manual describes the Cubefree package, a GAP 4 package for constructing groups of cubefree order; i.e., groups whose order is not divisible by any third power of a prime.
The groups of squarefree order are known for a long time: Hoelder Hol93 investigated them at the end of the 19th century. Taunt Tau55 has considered solvable groups of cubefree order, since he examined solvable groups with abelian Sylow subgroups. Cubefree groups in general are investigated firstly in Di05, DiEi05, and DiEi05add, and this package contains the implementation of the algorithms described there.
Some general approaches to construct groups of an arbitrarily given order are described in BeEia, BeEib, and BeEiO.
The main function of this package is a method to construct all groups of a given cubefree order up to isomorphism. The algorithm behind this function is described completely in Di05 and DiEi05. It is a refinement of the methods of the GrpConst package which are described in GrpConst.
This main function needs a method to construct up to conjugacy the solvable cubefree subgroups of GL(2,p) coprime to p. We split this construction into the construction of reducible and irreducible subgroups of GL(2,p). To determine the irreducible subgroups we use the method described in FlOB05 for which this package also contains an implementation. Alternatively, the IrredSol package Irredsol could be used for primes p ≤ 251.
The algorithm of FlOB05 requires a method to rewrite a matrix representation. We use and implement the method of GlHo97 for this purpose.
One can modify the construction algorithm for cubefree groups to a very
efficient algorithm to construct groups of squarefree order. This is already
done in the SmallGroups library. Thus for the construction of groups of squarefree order it is more practical to
use AllSmallGroups of the SmallGroups library.
A more detailed description of the implemented methods can be found in Chapter 2.
Chapter Installing and Loading the Cubefree Package explains how to install and load the Cubefree package.
In this section we give a brief survey about the main algorithm which is used to construct groups of cubefree order: the Frattini extension method. For a by far more detailed description we refer to the above references; e.g. see the online version of Di05.
Let G be a finite group. The Frattini subgroup Φ(G) is defined to be the intersection of all maximal subgroups of G. We say a group H is a Frattini extension by G if the Frattini factor H/Φ(H) is isomorphic to G. The Frattini factor of H is Frattini-free; i.e. it has a trivial Frattini subgroup. It is known that every prime divisor of |H| is also a divisor of |H/Φ(H)|. Thus the Frattini subgroup of a cubefree group has to be squarefree and, as it is nilpotent, it is a direct product of cyclic groups of prime order.
Hence in order to construct all groups of a given cubefree order n, say, one can, firstly, construct all Frattini-free groups of suitable orders and, secondly, compute all corresponding Frattini extensions of order n. A first fundamental result is that a group of cubefree order is either a solvable Frattini extension or a direct product of a PSL(2,r), r > 3 a prime, with a solvable Frattini extension. In particular, the simple groups of cubefree order are the groups PSL(2,r) with r > 3 a prime such that r±1 is cubefree. As a nilpotent group is the direct product of its Sylow subgroups, it is straightforward to compute all nilpotent groups of a given cubefree order.
Another important result is that for a cubefree solvable Frattini-free group there is exactly one isomorphism type of suitable Frattini extensions, which restricts the construction of cubefree groups to the determination of cubefree solvable Frattini-free groups. This uniqueness of Frattini extensions is the main reason why the Frattini extension method works so efficiently in the cubefree case.
In other words, there is a one-to-one correspondence between the solvable cubefree groups of order n and someFrattini-free groups of order dividing n. This allows to count the number of isomorphism types of cubefree groups of a given order without constructing Frattini extensions.
In the remaining part of this section we consider the construction of the solvable Frattini-free groups of a given cubefree order up to isomorphism. Such a group is a split extension over its socle; i.e. over the product of its minimal normal subgroups. Let F be a solvable Frattini-free group of cubefree order with socle S. Then S is a (cubefree) direct product of cyclic groups of prime order and F can be written as F=K\ltimes S where K ≤ Aut(S) is determined up to conjugacy. In particular, K is a subdirect product of certain cubefree subgroups of groups of the type GL(2,p) or Cp−1. Hence in order to determine all possible subgroups K one can determine all possible projections from such a subgroup into the direct factors of the types GL(2,p) and Cp−1, and then form all subdirect products having these projections. The construction of these subdirect products is one of the most time-consuming parts in the Frattini extension method for cubefree groups.
Cubefree manual